A292930 Triangle read by rows: T(n,k) (n>=1, 3<=k<=n+2) is the number of k-sequences of balls colored with at most n colors such that exactly three balls are the same color as some other ball in the sequence.

1, 2, 8, 3, 24, 60, 4, 48, 240, 480, 5, 80, 600, 2400, 4200, 6, 120, 1200, 7200, 25200, 40320, 7, 168, 2100, 16800, 88200, 282240, 423360, 8, 224, 3360, 33600, 235200, 1128960, 3386880, 4838400, 9, 288, 5040, 60480, 529200, 3386880, 15240960, 43545600, 59875200, 10, 360, 7200, 100800, 1058400, 8467200, 50803200, 217728000, 598752000, 798336000

Offset: 1

Author

Jeremy Dover, Sep 26 2017

Keywords

• nonn
• tabl

Comments

Note that the three matching balls are necessarily the same color.

Examples

			n=1 => AAA -> T(1,3)=1;
n=2 => AAA,BBB -> T(2,3)=2;
       AAAB,AABA,ABAA,BAAA,BBBA,BBAB,BABB,ABBB -> T(2,4)=8.
Triangle begins:
  1;
  2, 8;
  3, 24, 60;
  4, 48, 240, 480;
  5, 80, 600, 2400, 4200;
  ...
		

Crossrefs

Columns of table: T(n,3) = A000027(n), T(n,4) = A033996(n).

Other sequences in table: T(n,n+2) = A005990(n+1).

Programs

PARI

T(n, k) = binomial(k,3)*n!/(n+2-k)!;
tabl(nn) = for (n=1, nn, for (k=3, n+2, print1(T(n,k), ", ")); print()); \\ Michel Marcus, Sep 29 2017

Formula

T(n, k) = binomial(k,3)*n!/(n+2-k)!.